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Revision as of 09:17, 30 June 2014
Contents
 1 Mathematics
 1.1 Introductory
 1.2 Lojban numbers
 1.3 Signs and numerical punctuation
 1.4 Special numbers
 1.5 Simple infix expressions and equations
 1.6 Forethought operators (Polish notation, functions)
 1.7 Other useful selbri for mekso bridi
 1.8 Indefinite numbers
 1.9 Approximation and inexact numbers
 1.10 Nondecimal and compound bases
 1.11 Special mekso selbri
 1.12 Number questions
 1.13 Subscripts
 1.14 Infix operators revisited
 1.15 Vectors and matrices
 1.16 Reverse Polish notation
 1.17 Logical and nonlogical connectives within mekso
 1.18 Using Lojban resources within mekso
 1.19 Other uses of mekso
 1.20 Explicit operator precedence
 1.21 Miscellany
 1.22 Four score and seven: a mekso problem
 1.23 mekso selma'o summary
 1.24 Complete table of VUhU cmavo, with operand structures
 1.25 Complete table of PA cmavo: digits, punctuation, and other numbers.
 1.26 Table of MOI cmavo, with associated rafsi and place structures
Mathematics
Introductory
lojbau mekso ( “Lojbanic mathematicalexpression”) is the part of the Lojban language that is tailored for expressing statements of a mathematical character, or for adding numerical information to nonmathematical statements. Its formal design goals include:
 representing all the different forms of expression used by mathematicians in their normal modes of writing, so that a reader can unambiguously read off mathematical text as written with minimal effort and expect a listener to understand it;
 providing a vocabulary of commonly used mathematical terms which can readily be expanded to include newly coined words using the full resources of Lojban;
 permitting the formulation, both in writing and in speech, of unambiguous mathematical text;
 encompassing all forms of quantified expression found in natural languages, as well as encouraging greater precision in ordinary language situations than natural languages allow.
Goal 1 requires that mekso not be constrained to a single notation such as Polish notation or reverse Polish notation, but make provision for all forms, with the most commonly used forms the most easily used.
Goal 2 requires the provision of several conversion mechanisms, so that the boundary between mekso and full Lojban can be crossed from either side at many points.
Goal 3 is the most subtle. Written mathematical expression is culturally unambiguous, in the sense that mathematicians in all parts of the world understand the same written texts to have the same meanings. However, international mathematical notation does not prescribe unique forms. For example, the expression
 Example .1:
3x + 2y
contains omitted multiplication operators, but there are other possible interpretations for the strings
“3x” and “2y” than as mathematical multiplication. Therefore, the Lojban verbal (spoken and written) form of
Example .1 must not omit the multiplication operators.
The remainder of this chapter explains (in as much detail as is currently possible) the mekso system. This chapter is by intention complete as regards mekso components, but only suggestive about uses of those components – as of now, there has been no really comprehensive use made of mekso facilities, and many matters must await the test of usage to be fully clarified.
Lojban numbers
The following cmavo are discussed in this section:
 pa
 PA
 description
 re
 PA
 description
 ci
 PA
 description
 vo
 PA
 description
 mu
 PA
 description
 xa
 PA
 description
 ze
 PA
 description
 bi
 PA
 description
 so
 PA
 description
 no
 PA
 description
The simplest kind of mekso are numbers, which are cmavo or compound cmavo. There are cmavo for each of the 10 decimal digits, and numbers greater than 9 are made by stringing together the cmavo. Some examples:
 Example .2:
 pa re ci
one two three
123
 one hundred and twenty three
 Example .3:
 pa no
one zero
10
 ten
 Example .4:
 pa re ci vo mu xa ze bi so no
one two three four five six seven eight nine zero
1234567890
 one billion, two hundred and thirtyfour million, five hundred and sixtyseven thousand, eight hundred and ninety.
Therefore, there are no separate cmavo for “ten”, “hundred”, etc.
There is a pattern to the digit cmavo (except for no, 0) which is worth explaining. The cmavo from 1 to 5 end in the vowels a, e, i, o, u respectively; and the cmavo from 6 to 9 likewise end in the vowels a, e, i, and o respectively. None of the digit cmavo begin with the same consonant, to make them easy to tell apart in noisy environments.
Signs and numerical punctuation
The following cmavo are discussed in this section:
 ma'u
 PA
 description
 ni'u
 PA
 description
 pi
 PA
 description
 fi'u
 PA
 description
 ra'e
 PA
 description
 ce'i
 PA
 description
 ki'o
 PA
 description
A number can be given an explicit sign by the use of ma'u and ni'u, which are the positive and negative signs as distinct from the addition, subtraction, and negation operators. For example:
 Example .5:
 ni'u pa
negativesign 1
1
Grammatically, the signs are part of the number to which they are attached. It is also possible to use ma'u and ni'u by themselves as numbers; the meaning of these numbers is explained in Section .7.
Various numerical punctuation marks are likewise expressed by cmavo, as illustrated in the following examples:
 Example .6:
 ci pi pa vo pa mu
three point one four one five
3.1415
(In some cultures, a comma is used instead of a period in the symbolic version of Example .6; pi is still the Lojban representation for the decimal point.)
 Example .7:
 re fi'u ze
two fraction seven
2/7
Example .7 is the name of the number twosevenths; it is not the same as “the result of 2 divided by 7” in Lojban, although numerically these two are equal. If the denominator of the fraction is present but the numerator is not, the numerator is taken to be 1, thus expressing the reciprocal of the following number:
 Example .8:
 fi'u ze
fraction seven
1/7
 Example .9:
 pi ci mu ra'e pa vo re bi mu ze
point three five repeating one four two eight five seven
.35142857142857...
Note that the ra'e marks unambiguously where the repeating portion
“142857” begins.
 Example .10:
 ci mu ce'i
three five percent
35%
 Example .11:
 pa ki'o re ci vo ki'o mu xa ze
one comma two three four comma five six seven
1,234,567
(In some cultures, spaces are used in the symbolic representation of Example .11; ki'o is still the Lojban representation.)
It is also possible to have less than three digits between successive ki'o s, in which case zeros are assumed to have been elided:
 Example .12:
 pa ki'o re ci ki'o vo
one comma two three comma four
1,023,004
In the same way, ki'o can be used after
pi to divide fractions into groups of three:
 Example .13:
 pi ki'o re re
point comma two two
.022
 Example .14:
 pi pa ki'o pa re ki'o pa
point one comma one two comma one
.001012001
Special numbers
The following cmavo are discussed in this section:
 ci'i
 PA
 description
 ka'o
 PA
 description
 pai
 PA
 description
 te'o
 PA
 description
 fi'u
 PA
 description
The last cmavo is the same as the fraction sign cmavo: a fraction sign with neither numerator nor denominator represents the golden ratio.
Numbers can have any of these digit, punctuation, and specialnumber cmavo of Sections 2, 3, and 4 in any combination:
 Example .15:
 ma'u ci'i
+∞
 Example .16:
 ci ka'o re
 3i2 (a complex number equivalent to 3 + 2i)
Note that ka'o is both a special number (meaning
“i”) and a number punctuation mark (separating the real and the imaginary parts of a complex number).
 Example .17:
 ci'i no
 infinity zero
 ℵ<subscript>0</subscript> (a transfinite cardinal)
The special numbers pai and te'o are mathematically important, which is why they are given their own cmavo:
 Example .18:
 pai
 pi, π
 Example .19:
 te'o
 e
However, many combinations are as yet undefined:
 Example .20:
 pa pi re pi ci
 1.2.3
 Example .21:
 pa ni'u re
1 negativesign 2
Example .21 is not “1 minus 2”, which is represented by a different cmavo sequence altogether. It is a single number which has not been assigned a meaning. There are many such numbers which have no welldefined meaning; they may be used for experimental purposes or for future expansion of the Lojban number system.
It is possible, of course, that some of these “oddities” do have a meaningful use in some restricted area of mathematics. A mathematician appropriating these structures for specialized use needs to consider whether some other branch of mathematics would use the structure differently.
More information on numbers may be found in Section .7 to Section .11.
Simple infix expressions and equations
The following cmavo are discussed in this section:
 du
 GOhA
 description
 su'i
 VUhU
 description
 vu'u
 VUhU
 description
 pi'i
 VUhU
 description
 te'a
 VUhU
 description
 ny.
 BY
 description
“n”
 vei
 VEI
 description
 ve'o
 VEhO
 description
Let us begin at the beginning: one plus one equals two. In Lojban, that sentence translates to:
 Example .22:
 li pa su'i pa du li re
Thenumber one plus one equals thenumber two.
{{{1}}}
Example .22, a mekso sentence, is a regular Lojban bridi that exploits mekso features. du is the predicate meaning “x1 is mathematically equal to x2”. It is a cmavo for conciseness, but it has the same grammatical uses as any brivla. Outside mathematical contexts, du means “x1 is identical with x2” or “x1 is the same object as x2”.
The cmavo li is the number article. It is required whenever a sentence talks about numbers as numbers, as opposed to using numbers to quantify things. For example:
 Example .23:
 le ci prenu
 the three persons
requires no li article, because the
ci is being used to specify the number of prenu. However, the sentence
 Example .24:
 levi sfani cu grake li ci
This fly massesingrams thenumber three.
 This fly has a mass of 3 grams.
requires li because ci is being used as a sumti. Note that this is the way in which measurements are stated in Lojban: all the predicates for units of length, mass, temperature, and so on have the measured object as the first place and a number as the second place. Using
li for le in Example .23 would produce
 Example .25:
 li ci prenu
Thenumber 3 isaperson.
which is grammatical but nonsensical: numbers are not persons.
The cmavo su'i belongs to selma'o VUhU, which is composed of mathematical operators, and means
“addition”. As mentioned before, it is distinct from ma'u which means the positive sign as an indication of a positive number:
 Example .26:
 li ma'u pa su'i ni'u pa du li no
Thenumber positivesign one plus negativesign one equals thenumber zero.
{{{1}}}
Of course, it is legal to have complex mekso on both sides of du:
 Example .27:
 li mu su'i pa du li ci su'i ci
Thenumber five plus one equals thenumber three plus three.
{{{1}}}
Why don't we say
 li mu su'i li pa rather than just
 li mu su'i pa? The answer is that VUhU operators connect mekso operands (numbers, in
Example .27), not general sumti.
li is used to make the entire mekso into a sumti, which then plays the roles applicable to other sumti: in Example .27, filling the places of a bridi
By default, Lojban mathematics is like simple calculator mathematics: there is no notion of
“operator precedence”. Consider the following example, where
pi'i means “times”, the multiplication operator:
 Example .28:
 li ci su'i vo pi'i mu du li reci
Thenumber three plus four times five equals thenumber twothree.
{{{1}}}
Is the Lojban version of Example .28 true? No! “3 + 4 × 5” is indeed 23, because the usual conventions of mathematics state that multiplication takes precedence over addition; that is, the multiplication
“4 × 5” is done first, giving 20, and only then the addition “3 + 20”. But VUhU operators by default are done left to right, like other Lojban grouping, and so a truthful bridi would be:
 Example .29:
 li ci su'i vo pi'i mu du li cimu
Thenumber three plus four times five equals thenumber threefive.
{{{1}}}
Here we calculate 3 + 4 first, giving 7, and then calculate 7 × 5 second, leading to the result 35. While possessing the advantage of simplicity, this result violates the design goal of matching the standards of mathematics. What can be done?
There are three solutions, all of which will probably be used to some degree. The first solution is to ignore the problem. People will say
 li ci su'i vo pi'i mu and mean 23 by it, because the notion that multiplication takes precedence over addition is too deeply ingrained to be eradicated by Lojban parsing, which totally ignores semantics. This convention essentially allows semantics to dominate syntax in this one area.
(Why not hardwire the precedences into the grammar, as is done in computer programming languages? Essentially because there are too many operators, known and unknown, with levels of precedence that vary according to usage. The programming language 'C' has 13 levels of precedence, and its list of operators is not even extensible. For Lojban this approach is just not practical. In addition, hardwired precedence could not be overridden in mathematical systems such as spreadsheets where the conventions are different.)
The second solution is to use explicit means to specify the precedence of operators. This approach is fully general, but clumsy, and will be explained in Section .19.
The third solution is simple but not very general. When an operator is prefixed with the cmavo bi'e (of selma'o BIhE), it becomes automatically of higher precedence than other operators not so prefixed. Thus,
 Example .30:
 li ci su'i vo bi'e pi'i mu du li reci
Thenumber three plus fourtimesfive equals thenumber twothree.
{{{1}}}
is a truthful Lojban bridi. If more than one operator has a bi'e prefix, grouping is from the right; multiple
bi'e prefixes on a single operator are not allowed.
In addition, of course, Lojban has the mathematical parentheses vei and ve'o, which can be used just like their written equivalents “(” and “)” to group expressions in any way desired:
 Example .31:
 li vei ny. su'i pa ve'o pi'i vei ny. su'i pa [ve'o] du li ny. [bi'e] te'a re su'i re bi'e pi'i ny. su'i pa
Thenumber (n plus one) times (n plus one) equals thenumber npowertwo plus twotimes n plus 1.
{{{1}}}
There are several new usages in Example .31: te'a means
“raised to the power”, and we also see the use of the lerfu word ny, representing the letter “n”. In mekso, letters stand for just what they do in ordinary mathematics: variables. The parser will accept a string of lerfu words (called a “lerfu string”) as the equivalent of a single lerfu word, in agreement with computerscience conventions; “abc” is a single variable, not the equivalent of “a × b × c”. (Of course, a local convention could state that the value of a variable like “abc”, with a multilerfu name, was equal to the values of the variables “a”, “b”, and “c” multiplied together.)
The explicit operator pi'i is required in the Lojban verbal form whereas multiplication is implicit in the symbolic form. Note that
ve'o (the right parenthesis) is an elidable terminator: the first use of it in Example .31 is required, but the second use (marked by square brackets) could be elided. Additionally, the first
bi'e (also marked by square brackets) is not necessary to get the proper grouping, but it is included here for symmetry with the other one.
Forethought operators (Polish notation, functions)
The following cmavo are discussed in this section:
 boi
 BOI
 description
 va'a
 VUhU
 description
 pe'o
 PEhO
 description
 ku'e
 KUhE
 description
 py.
 BY
 description
 xy.
 BY
 description
 zy.
 BY
 description
 ma'o
 MAhO
 description
 fy.
 BY
 description
The infix form explained so far is reasonable for many purposes, but it is limited and rigid. It works smoothly only where all operators have exactly two operands, and where precedences can either be assumed from context or are limited to just two levels, with some help from parentheses.
But there are many operators which do not have two operands, or which have a variable number of operands. The preferred form of expression in such cases is the use of “forethought operators”, also known as Polish notation. In this style of writing mathematics, the operator comes first and the operands afterwards:
 Example .32:
 li su'i paboi reboi ci[boi] du li xa
Thenumber thesumof one two three equals thenumber six.
{{{1}}}
Note that the normally elidable number terminator boi is required after pa and re because otherwise the reading would be
 pareci= 123. It is not required after
ci but is inserted here in brackets for the sake of symmetry. The only time boi is required is, as in Example .32, when there are two consecutive numbers or lerfu strings.
Forethought mekso can use any number of operands, in Example .32, three. How do we know how many operands there are in ambiguous circumstances? The usual Lojban solution is employed: an elidable terminator, namely ku'e. Here is an example:
 Example .33:
 li py. su'i va'a ny. ku'e su'i zy du li xy.
Thenumber
“p” plus negativeof(
“n”) plus
“z” equals thenumber
“x”.
{{{1}}}
where we know that va'a is a forethought operator because there is no operand preceding it.
va'a is the numerical negation operator, of selma'o VUhU. In contrast,
vu'u is not used for numerical negation, but only for subtraction, as it always has two or more operands. Do not confuse va'a and vu'u, which are operators, with ni'u, which is part of a number.
In Example .33, the operator va'a and the terminator ku'e serve in effect as parentheses. (The regular parentheses vei and ve'o are NOT used for this purpose.) If the ku'e were omitted, the
 su'i zy would be swallowed up by the
va'a forethought operator, which would then appear to have two operands, ny and
 su'i zy., where the latter is also a forethought expression.
Forethought mekso is also useful for matching standard functional notation. How do we represent
“z = f(x)”? The answer is:
 Example .34:
 li zy du li ma'o fy.boi xy.
Thenumber z equals thenumber theoperator f x.
{{{1}}}
Again, no parentheses are used. The construct
 ma'o fy.boi is the equivalent of an operator, and appears in forethought here (although it could also be used as a regular infix operator). In mathematics, letters sometimes mean functions and sometimes mean variables, with only the context to tell which. Lojban chooses to accept the variable interpretation as the default, and uses the special flag
ma'o to mark a lerfu string as an operator. The cmavo xy. and zy. are variables, but fy. is an operator (a function) because ma'o marks it as such. The boi is required because otherwise the xy. would look like part of the operator name. (The use of ma'o can be generalized from lerfu strings to any mekso operand: See Section .20.)
When using forethought mekso, the optional marker pe'o may be placed in front of the operator. This usage can help avoid confusion by providing clearly marked
pe'o and
ku'e pairs to delimit the operand list. Example .32 to Example .34, respectively, with explicit pe'o and
ku'e:
 Example .35:
 li pe'o su'i paboi reboi ciboi ku'e du li xa
 Example .36:
 li py. su'i pe'o va'a ny. ku'e su'i zy du li xy.
 Example .37:
 li zy du li pe'o ma'o fy.boi xy. ku'e
Note: When using forethought mekso, be sure that the operands really are operands: they cannot contain regular infix expressions unless parenthesized with
vei and ve'o. An earlier version of the complex Example .119 came to grief because I forgot this rule.
Other useful selbri for mekso bridi
So far our examples have been isolated mekso (it is legal to have a bare mekso as a sentence in Lojban) and equation bridi involving du. What about inequalities such as “x < 5”? The answer is to use a bridi with an appropriate selbri, thus:
 Example .38:
 li xy. mleca li mu
Thenumber x islessthan thenumber 5.
Here is a partial list of selbri useful in mathematical bridi:
 du
<listitem>:x1 is identical to x2, x3, x4, ...
 dunli
<listitem>:x1 is equal/congruent to x2 in/on property/quality/dimension/quantity x3
 mleca
<listitem>:x1 is less than x2
 zmadu
<listitem>:x1 is greater than x2
 dubjavme'a
<listitem>:x1 is less than or equal to x2 [du ja mleca, equal or less]
 dubjavmau
<listitem>:x1 is greater than or equal to x2 [du ja zmadu, equal or greater]
 tamdu'i
<listitem>:x1 is similar to x2 [tarmi dunli, shapeequal]
 turdu'i
<listitem>:x1 is isomorphic to x2 [stura dunli, structureequal]
 cmima
<listitem>:x1 is a member of set x2
 gripau
<listitem>:x1 is a subset of set x2 [girzu pagbu, setpart]
 na'ujbi
<listitem>:x1 is approximately equal to x2 [namcu jibni, numbernear]
 terci'e
<listitem>:x1 is a component with function x2 of system x3
Note the difference between dunli and
du; dunli has a third place that specifies the kind of equality that is meant.
du refers to actual identity, and can have any number of places:
 Example .39:
 py. du xy.boi zy.
“p” isidenticalto
“x”
“z”
{{{1}}}
Lojban bridi can have only one predicate, so the du is not repeated.
Any of these selbri may usefully be prefixed with na, the contradictory negation cmavo, to indicate that the relation is false:
 Example .40:
 li re su'i re na du li mu
thenumber 2 + 2 isnot equalto thenumber 5.
2 + 2 ≠ 5
As usual in Lojban, negated bridi say what is false, and do not say anything about what might be true.
Indefinite numbers
The following cmavo are discussed in this section:
 ro
 PA
 description
 PA
 description
 PA
 description
 PA
 description
 PA
 description
 PA
 description
 PA
 description
 da'a
 PA
 description
 PA
 description
 PA
 description
 PA
 description
Not all the cmavo of PA represent numbers in the usual mathematical sense. For example, the cmavo ro means “all” or “each”. This number does not have a definite value in the abstract:
 li ro is undefined. But when used to count or quantify something, the parallel between
ro and pa is clearer:
 Example .41:
 mi catlu pa prenu
I lookat one person
 Example .42:
 mi catlu ro prenu
I lookat all persons
Example .41 might be true, whereas Example .42 is almost certainly false.
The cmavo so'a,
so'e,
so'i,
so'o, and
so'u represent a set of indefinite numbers less than
ro. As you go down an alphabetical list, the magnitude decreases:
 Example .43:
 mi catlu so'a prenu
I lookat almostall persons
 Example .44:
 mi catlu so'e prenu
I lookat most persons
 Example .45:
 mi catlu so'i prenu
I lookat many persons
 Example .46:
 mi catlu so'o prenu
I lookat several persons
 Example .47:
 mi catlu so'u prenu
I lookat afew persons
The English equivalents are only rough: the cmavo provide space for up to five indefinite numbers between
ro and no, with a builtin ordering. In particular, so'e does not mean
“most” in the sense of “a majority” or “more than half”.
Each of these numbers, plus ro, may be prefixed with pi (the decimal point) in order to make a fractional form which represents part of a whole rather than some elements of a totality.
piro therefore means “the whole of”:
 Example .48:
 mi citka piro lei nanba
I eat thewholeof themassof bread
Similarly, piso'a means “almost the whole of”; and so on down to piso'u, “a tiny part of”. These numbers are particularly appropriate with masses, which are usually measured rather than counted, as Example .48 shows.
In addition to these cmavo, there is no'o, meaning
“the typical value”, and
 pino'o, meaning
“the typical portion”: Sometimes no'o can be translated
“the average value”, but the average in question is not, in general, a mathematical mean, median, or mode; these would be more appropriately represented by operators.
 Example .49:
 mi catlu no'o prenu
I lookat atypicalnumberof persons
 Example .50:
 mi citka pino'o lei nanba
I eat atypicalamountof themassof bread.
da'a is a related cmavo meaning “all but”:
 Example .51:
 mi catlu da'a re prenu
I lookat allbut two persons
 Example .52:
 mi catlu da'a so'u prenu
I lookat allbut afew persons
Example .52 is similar in meaning to Example .43.
If no number follows da'a, then pa is assumed; da'a by itself means “all but one”, or in ordinal contexts “all but the last”:
 Example .53:
 ro ratcu ka'e citka da'a ratcu
All rats can eat allbutone rats.
 All rats can eat all other rats.
(The use of da'a means that Example .53 does not require that all rats can eat themselves, but does allow it. Each rat has one rat it cannot eat, but that one might be some rat other than itself. Context often dictates that
“itself” is, indeed, the “other” rat.)
As mentioned in Section .2, ma'u and ni'u are also legal numbers, and they mean “some positive number” and “some negative number” respectively.
 Example .54:
 li ci vu'u re du li ma'u
thenumber 3 − 2 = somepositivenumber
 Example .55:
 li ci vu'u vo du li ni'u
thenumber 3 − 4 = somenegativenumber
 Example .56:
 mi ponse ma'u rupnu
I possess apositivenumberof currencyunits.
All of the numbers discussed so far are objective, even if indefinite. If there are exactly six superpowers ( rairgugde, “superlativestates”) in the world, then
 ro rairgugde means the same as
 xa rairgugde. It is often useful, however, to express subjective indefinite values. The cmavo
rau (enough),
du'e (too many), and
mo'a (too few) are then appropriate:
 Example .57:
 mi ponse rau rupnu
I possess enough currencyunits.
Like the so'aseries,
rau,
du'e, and
mo'a can be preceded by
pi; for example,
 pirau means
“a sufficient part of.”
Another possibility is that of combining definite and indefinite numbers into a single number. This usage implies that the two kinds of numbers have the same value in the given context:
 Example .58:
 mi viska le rore gerku
I saw the allof/two dogs.
 I saw both dogs.
 Example .59:
 mi speni so'ici prenu
I ammarriedto many/three persons.
 I am married to three persons (which is “many” in the circumstances).
Example .59 assumes a mostly monogamous culture by stating that three is “many”.
Approximation and inexact numbers
The following cmavo are discussed in this section:
 ji'i
 PA
 description
 su'e
 PA
 description
 su'o
 PA
 description
 me'i
 PA
 description
 za'u
 PA
 description
The cmavo ji'i (of selma'o PA) is used in several ways to indicate approximate or rounded numbers. If it appears at the beginning of a number, the whole number is approximate:
 Example .60:
 ji'i vo no
approximation four zero
 approximately 40
If ji'i appears in the middle of a number, all the digits following it are approximate:
 Example .61:
 vo no ji'i mu no
four zero approximation five zero
<natlang>roughly 4050 (where the “four thousand” is exact, but the “fifty” is approximate)</natlang>
If ji'i appears at the end of a number, it indicates that the number has been rounded. In addition, it can then be followed by a sign cmavo ( ma'u or ni'u), which indicate truncation towards positive or negative infinity respectively.
 Example .62:
 re pi ze re ji'i
two point seven two approximation
 2.72 (rounded)
 Example .63:
 re pi ze re ji'i ma'u
two point seven two approximation positivesign
 2.72 (rounded up)
 Example .64:
 re pi ze pa ji'i ni'u
two point seven one approximation negativesign
 2.71 (rounded down)
Example .62 through Example .64 are all approximations to te'o (exponential e). ji'i can also appear by itself, in which case it means “approximately the typical value in this context”.
The four cmavo su'e,
su'o, me'i, and
za'u, also of selma'o PA, express inexact numbers with upper or lower bounds:
 Example .65:
 mi catlu su'e re prenu
I lookat atmost two persons
 Example .66:
 mi catlu su'o re prenu
I lookat atleast two persons
 Example .67:
 mi catlu me'i re prenu
I lookat lessthan two persons
 Example .68:
 mi catlu za'u re prenu
I lookat morethan two persons
Each of these is a subtly different claim: Example .66 is true of two or any greater number, whereas Example .68 requires three persons or more. Likewise, Example .65 refers to zero, one, or two; Example .67 to zero or one. (Of course, when the context allows numbers other than nonnegative integers,
 me'i re can be any number less than 2, and likewise with the other cases.) The exact quantifier,
“exactly 2, neither more nor less” is just re. Note that
 su'ore is the exact Lojban equivalent of English plurals.
If no number follows one of these cmavo, pa is understood: therefore,
 Example .69:
 mi catlu su'o prenu
I lookat atleast [one] person
is a meaningful claim.
Like the numbers in Section .7, all of these cmavo may be preceded by pi to make the corresponding quantifiers for part of a whole. For example, pisu'o means
“at least some part of”. The quantifiers ro, su'o, piro, and pisu'o are particularly important in Lojban, as they are implicitly used in the descriptions introduced by the cmavo of selma'o LA and LE, as explained in
Section . Descriptions in general are outside the scope of this chapter.
Nondecimal and compound bases
The following cmavo are discussed in this section:
 ju'u
 VUhU
 description
 dau
 PA
 description
 fei
 PA
 description
 gai
 PA
 description
 jau
 PA
 description
 rei
 PA
 description
 vai
 PA
 description
 pi'e
 PA
 description
In normal contexts, Lojban assumes that all numbers are expressed in the decimal (base 10) system. However, other bases are possible, and may be appropriate in particular circumstances.
To specify a number in a particular base, the VUhU operator ju'u is suitable:
 Example .70:
 li pa no pa no ju'u re du li pa no
Thenumber 1010 base 2 equals thenumber 10.
Here, the final
 pa no is assumed to be base 10, as usual; so is the base specification. (The base may also be changed permanently by a metalinguistic specification; no standard way of doing so has as yet been worked out.)
Lojban has digits for representing bases up to 16, because 16 is a base often used in computer applications. In English, it is customary to use the letters AF as the base 16 digits equivalent to the numbers ten through fifteen. In Lojban, this ambiguity is avoided:
 Example .71:
 li daufeigai ju'u paxa du li rezevobi
Thenumber ABC base 16 equals thenumber 2748.
 Example .72:
 li jaureivai ju'u paxa du li cimuxaze
Thenumber DEF base 16 equals thenumber 3567.
Note the pattern in the cmavo: the diphthongs au, ei, ai are used twice in the same order. The digits for A to D use consonants different from those used in the decimal digit cmavo; E and F unfortunately overlap 2 and 4 – there was simply not enough available cmavo space to make a full differentiation possible. The cmavo are also in alphabetical order.
The base point
pi is used in nondecimal bases just as in base 10:
 Example .73:
 li vai pi bi ju'u paxa du li pamu pi mu
Thenumber F.8 base 16 equals thenumber 15.5.
Since ju'u is an operator of selma'o VUhU, it is grammatical to use any operand as the left argument. Semantically, however, it is undefined to use anything but a numeral string on the left. The reason for making ju'u an operator is to allow reference to a base which is not a constant.
There are some numerical values that require a “base” that varies from digit to digit. For example, times represented in hours, minutes, and seconds have, in effect, three
“digits”: the first is base 24, the second and third are base 60. To express such numbers, the compound base separator
pi'e is used:
 Example .74:
 ci pi'e rere pi'e vono
3:22:40
Each digit sequence separated by instances of pi'e is expressed in decimal notation, but the number as a whole is not decimal and can only be added and subtracted by special rules:
 Example .75:
 li ci pi'e rere pi'e vono su'i pi'e ci pi'e cici du li ci pi'e rexa pi'e paci
Thenumber 3:22:40 plus :3:33 equals thenumber 3:26:13.
{{{1}}}
Of course, only context tells you that the first part of the numbers in Example .74 and Example .75 is hours, the second minutes, and the third seconds.
The same mechanism using pi'e can be used to express numbers which have a base larger than 16. For example, base20 Mayan mathematics might use digits from
no to paso, each separated by pi'e:
 Example .76:
 li pa pi'e re pi'e ci ju'u reno du li vovoci
thenumber 1;2;3 base 20 equals thenumber 443
Carefully note the difference between:
 Example .77:
 pano ju'u reno
thedigit10 base 20
which is equal to ten, and:
 Example .78:
 pa pi'e no ju'u reno
1;0 base 20
which is equal to twenty.
Both pi and pi'e can be used to express largebase fractions:
 Example .79:
 li pa pi'e vo pi ze ju'u reno du li re vo pi ci mu
Thenumber 1;4.7 base 20 equals thenumber 24.35.
pi'e is also used where the base of each digit is vague, as in the numbering of the examples in this chapter:
 Example .80:
 dei jufra panopi'epapamoi
Thisutterance isasentencetypeof 10;11ththing.
 This is Sentence 10.11.
Special mekso selbri
The following cmavo are discussed in this section:
 mei
 MOI
 description
 moi
 MOI
 description
 MOI
 description
 MOI
 description
 MOI
 description
 me
 ME
 description
 me'u
 MEhU
 description
Lojban possesses a special category of selbri which are based on mekso. The simplest kind of such selbri are made by suffixing a member of selma'o MOI to a number. There are five members of MOI, each of which serves to create numberbased selbri with specific place structures.
The cmavo mei creates cardinal selbri. The basic place structure is:
 x1 is a mass formed from the set x2 of n members, one or more of which is/are x3
A cardinal selbri interrelates a set with a given number of members, the mass formed from that set, and the individuals which make the set up. The mass argument is placed first as a matter of convenience, not logical necessity.
Some examples:
 Example .81:
 lei mi ratcu cu cimei
ThoseIdescribeasthemassof my rats areathreesome.
My rats are three.
 I have three rats.
Here, the mass of my rats is said to have three components; that is, I have three rats.
Another example, with one element this time:
 Example .82:
 mi poi pamei cu cusku dei
I who amanindividual express thissentence.
In Example .82, mi refers to a mass, “the mass consisting of me”. Personal pronouns are vague between masses, sets, and individuals.
However, when the number expressed before mei is an objective indefinite number of the kind explained in Section .7, a slightly different place structure is required:
 x1 is a mass formed from a set x2 of n members, one or more of which is/are x3, measured relative to the set x4.
An example:
 Example .83:
 lei ratcu poi zvati le panka cu so'umei lo'i ratcu
Themassof rats which arein the park are afewsome withrespectto thesetof rats.
 The rats in the park are a small number of all the rats there are.
In Example .83, the x2 and x3 places are vacant, and the x4 place is filled by
 lo'i ratcu, which (because no quantifiers are explicitly given) means
“the whole of the set of all those things which are rats”, or simply “the set of all rats.”
 Example .84:
 le'i ratcu poi zvati le panka cu se so'imei
Thesetof rats whichare in the park isa manysome.
 There are many rats in the park.
In Example .84, the conversion cmavo se swaps the x1 and the x2 places, so that the new x1 is the set. The x4 set is unspecified, so the implication is that the rats are “many” with respect to some unspecified comparison set.
More explanations about the interrelationship of sets, masses, and individuals can be found in Section .
The cmavo moi creates ordinal selbri. The place structure is:
 x1 is the (n)th member of set x2 when ordered by rule x3
Some examples:
 Example .85:
 ti pamoi le'i mi ratcu
Thisone is the firstof the rats associatedwith me.
 This is my first rat.
 Example .86:
 ta romoi le'i mi ratcu
That istheallthof the rats associatedwith me.
 That is my last rat.
 Example .87:
 mi raumoi le velskina porsi
I amenoughthin the movieaudience sequence
I am enoughth in the movie line.
Example .87 means, in the appropriate context, that my position in line is sufficiently far to the front that I will get a seat for the movie.
The cmavo si'e creates portion selbri. The place structure is:
 x1 is an (n)th portion of mass x2
Some examples:
 Example .88:
 levi sanmi cu fi'ucisi'e lei mi djedi cidja
Thishere meal isaslashthreeportionof my dayfood.
 This meal is onethird of my daily food.
The cmavo cu'o creates probability selbri. The place structure is:
 event x1 has probability (n) of occurring under conditions x2
The number must be between 0 and 1 inclusive. For example:
 Example .89:
 le nu lo sicni cu sedja'o cu pimucu'o
The event of a coin being a headdisplayer has probability .5.
The cmavo va'e creates a scale selbri. The place structure is:
 x1 is at scale position (n) on the scale x2
If the scale is granular rather than continuous, a form like
 cifi'uxa (3/6) may be used; in this case, 3/6 is not the same as 1/2, because the third position on a scale of six positions is not the same as the first position on a scale of two positions. Here is an example:
 Example .90:
 le vi rozgu cu sofi'upanova'e xunre
This rose is 9/10scale red.
This rose is 9 out of 10 on the scale of redness.
 This rose is very red.
When the quantifier preceding any MOI cmavo includes the subjective numbers rau,
du'e, or
mo'a (enough, too many, too few) then an additional place is added for
“by standard”. For example:
 Example .91:
 lei ratcu poi zvati le panka cu du'emei fo mi
Themassof rats whichare in the park are toomany bystandard me.
 There are too many rats in the park for me.
The extra place (which for
 mei is the x4 place labeled by
fo) is provided rather than using a BAI tag such as ma'i because a specification of the standard for judgment is essential to the meaning of subjective words like
“enough”.
This place is not normally explicit when using one of the subjective numbers directly as a number. Therefore,
 du'e ratcu means
“too many rats” without specifying any standard.
It is also grammatical to substitute a lerfu string for a number:
 Example .92:
 ta ny.moi le'i mi ratcu
That isnthof thesetof my rats.
 That is my nth rat.
More complex mekso cannot be placed directly in front of MOI, due to the resulting grammatical ambiguities. Instead, a somewhat artificial form of expression is required.
The cmavo me (of selma'o ME) has the function of making a sumti into a selbri. A whole me construction can have a member of MOI added to the end to create a complex mekso selbri:
 Example .93:
 ta me li ny. su'i pa me'u moi le'i mi ratcu
That is thenumber n plus onethof thesetof my rats.
That is my (n+1)th rat.
Here the mekso
 ny. su'i pa is made into a sumti (with
li) and then changed into a mekso selbri with me and
 me'u moi. The elidable terminator
me'u is required here in order to keep the pa and the moi separate; otherwise, the parser will combine them into the compound pamoi and reject the sentence as ungrammatical.
It is perfectly possible to use nonnumerical sumti after me and before a member of MOI, producing strange results indeed:
 Example .94:
 le nu mi nolraitru cu me le'e snime bolci be vi la xel. cu'o
The eventof me beinganoblysuperlativeruler hasthestereotypical snow typeofball at Hell probability.
 I have a snowball's chance in Hell of being king.
Note: the elidable terminator boi is not used between a number and a member of MOI. As a result, the me'u in Example .93 could also be replaced by a boi, which would serve the same function of preventing the pa and moi from joining into a compound.
Number questions
The following cmavo is discussed in this section:
 PA
 description
The cmavo xo, a member of selma'o PA, is used to ask questions whose answers are numbers. Like most Lojban question words, it fills the blank where the answer should go. (See
Section for more on Lojban questions.)
 Example .95:
 li re su'i re du li xo
Thenumber 2 plus 2 equals thenumber what?
 What is 2 + 2?
 Example .96:
 le xomoi prenu cu darxi do
The whatnumberth person hit you?
Which person [as in a police lineup] hit you?
xo can also be combined with other digits to ask questions whose answers are already partly specified. This ability could be very useful in writing tests of elementary arithmetical knowledge:
 Example .97:
 li remu pi'i xa du li paxono
Thenumber 25 times 6 equals thenumber 1?0
to which the correct reply would be mu, or 5. The ability to utter bare numbers as grammatical Lojban sentences is primarily intended for giving answers to xo questions. (Another use, obviously, is for counting off physical objects one by one.)
Subscripts
The following cmavo is discussed in this section:
 xi
 XI
 description
Subscripting is a general Lojban feature, not used only in mekso; there are many things that can logically be subscripted, and grammatically a subscript is a free modifier, usable almost anywhere. In particular, of course, mekso variables (lerfu strings) can be subscripted:
 Example .98:
 li xy.boixici du li xy.boixipa su'i xy.boixire
Thenumber xsub3 equals thenumber xsub1 plus xsub2.
{{{1}}}
Subscripts always begin with the flag xi (of selma'o XI). xi may be followed by a number, a lerfu string, or a general mekso expression in parentheses:
 Example .99:
 xy.boixino
x<subscript>0</subscript>
 Example .100:
 xy.boixiny.
x<subscript>n</subscript>
 Example .101:
 xy.boixi vei ny. su'i pa [ve'o]
x<subscript>(n+1)</subscript>
Note that subscripts attached directly to lerfu words (variables) generally need a boi terminating the variable. Free modifiers, of which subscripts are one variety, generally require the explicit presence of an otherwise elidable terminator.
There is no standard way of handling superscripts (other than those used as exponents) or for subscripts or superscripts that come before the main expression. If necessary, further cmavo could be assigned to selma'o XI for these purposes.
The elidable terminator for a subscript is that for a general number or lerfu string, namely boi. By convention, a subscript following another subscript is taken to be a subsubscript:
 Example .102:
 xy.boi xi by.boi xi vo
x<subscript>b<subscript>4</subscript></subscript>
See Example .123 for the standard method of specifying multiple subscripts on a single object.
More information on the uses of subscripts may be found in Section .
Infix operators revisited
The following cmavo are discussed in this section:
 PA
 description
 ge'a
 VUhU
 description
 gei
 VUhU
 description
The infix operators presented so far have always had exactly two operands, and for more or fewer operands forethought notation has been required. However, it is possible to use an operator in infix style even though it has more or fewer than two operands, through the use of a pair of tricks: the null operand
tu'o and the null operator
ge'a. The first is suitable when there are too few operands, the second when there are too many. For example, suppose we wanted to express the numerical negation operator
va'a in infix form. We would use:
 Example .103:
 li tu'o va'a ny. du li no vu'u ny.
Thenumber (null) additiveinverse n equals thenumber zero minus n.
{{{1}}}
The tu'o fulfills the grammatical requirement for a left operand for the infix use of
va'a, even though semantically none is needed or wanted.
Finding a suitable example of ge'a requires exhibiting a ternary operator, and ternary operators are not common. The operator gei, however, has both a binary and a ternary use. As a binary operator, it provides a terse representation of scientific (also called “exponential”) notation. The first operand of gei is the exponent, and the second operand is the mantissa or fraction:
 Example .104:
 li cinonoki'oki'o du li bi gei ci
Thenumber threezerozerocommacomma equals thenumber eight scientific three.
{{{1}}}
Why are the arguments to gei in reverse order from the conventional symbolic notation? So that gei can be used in forethought to allow easy specification of a large (or small) imprecise number:
 Example .105:
 gei reno
(scientific) twozero
10 ^{20}
Note, however, that although 10 is far and away the most common exponent base, it is not the only possible one. The third operand of gei, therefore, is the base, with 10 as the default value. Most computers internally store socalled “floatingpoint” numbers using 2 as the exponent base. (This has nothing to do with the fact that computers also represent all integers in base 2; the IBM 360 series used an exponent base of 16 for floating point, although each component of the number was expressed in base 2.) Here is a computer floatingpoint number with a value of 40:
 Example .106:
 papano bi'eju'u re gei pipanopano bi'eju'u re ge'a re
(oneonezero base 2) scientific (pointonezeroonezero base 2) withbase 2
.1010<subscript>2</subscript> x 2^{110<subscript>2</subscript>}
Vectors and matrices
The following cmavo are discussed in this section:
 jo'i
 JOhI
 description
 te'u
 TEhU
 description
 VUhU
 description
 VUhU
 description
A mathematical vector is a list of numbers, and a mathematical matrix is a table of numbers. Lojban considers matrices to be built up out of vectors, which are in turn built up out of operands.
jo'i, the only cmavo of selma'o JOhI, is the vector indicator: it has a syntax reminiscent of a forethought operator, but has very high precedence. The components must be simple operands rather than full expressions (unless parenthesized). A vector can have any number of components;
te'u is the elidable terminator. An example:
 Example .107:
 li jo'i paboi reboi te'u su'i jo'i ciboi voboi du li jo'i voboi xaboi
Thenumber array (one, two) plus array (three, four) equals thenumber array (four, six).
{{{1}}}
Vectors can be combined into matrices using either pi'a, the matrix row operator, or
sa'i, the matrix column operator. The first combines vectors representing rows of the matrix, and the second combines vectors representing columns of the matrix. Both of them allow any number of arguments: additional arguments are tacked on with the null operator
ge'a.
Therefore, the “magic square” matrix
<tab class=wikitable header=true> 8 1 6 3 5 7 4 9 2 </tab> can be represented either as:
 Example .108:
 jo'i biboi paboi xa pi'a jo'i ciboi muboi ze ge'a jo'i voboi soboi re
thevector (8 1 6) matrixrow thevector (3 5 7), thevector (4 9 2)
or as
 Example .109:
 jo'i biboi ciboi vo sa'i jo'i paboi muboi so ge'a jo'i xaboi zeboi re
thevector (8 3 4) matrixcolumn thevector (1 5 9), thevector (6 7 2)
The regular mekso operators can be applied to vectors and to matrices, since grammatically both of these are expressions. It is usually necessary to parenthesize matrices when used with operators in order to avoid incorrect groupings. There are no VUhU operators for the matrix operators of inner or outer products, but appropriate operators can be created using a suitable symbolic lerfu word or string prefixed by ma'o.
Matrices of more than two dimensions can be built up using either pi'a or
sa'i with an appropriate subscript numbering the dimension. When subscripted, there is no difference between
pi'a and
sa'i.
Reverse Polish notation
The following cmavo is discussed in this section:
 FUhA
 description
So far, the Lojban notational conventions have mapped fairly familiar kinds of mathematical discourse. The use of forethought operators may have seemed odd when applied to “+”, but when applied to “f” they appear as the usual functional notation. Now comes a sharp break. Reverse Polish (RP) notation represents something completely different; even mathematicians don't use it much. (The only common uses of RP, in fact, are in some kinds of calculators and in the implementation of some programming languages.)
In RP notation, the operator follows the operands. (Polish notation, where the operator precedes its operands, is another name for forethought mekso of the kind explained in Section .5.) The number of operands per operator is always fixed. No parentheses are required or permitted. In Lojban, RP notation is always explicitly marked by a fu'a at the beginning of the expression; there is no terminator. Here is a simple example:
 Example .110:
 li fu'a reboi ci su'i du li mu
thenumber (RP!) two, three, plus equals thenumber five.
The operands are re and ci; the operator is su'i.
Here is a more complex example:
 Example .111:
 li fu'a reboi ci pi'i voboi mu pi'i su'i du li rexa
thenumber (RP!) (two, three, times), (four, five, times), plus equals thenumber twosix
Here the operands of the first pi'i are re and ci; the operands of the second pi'i are vo and mu (with boi inserted where needed), and the operands of the su'i are
 reboi ci pi'i, or 6, and
 voboi mu pi'i, or 20. As you can see, it is easy to get lost in the world of reverse Polish notation; on the other hand, it is especially easy for a mechanical listener (who has a deep mental stack and doesn't get lost) to comprehend.
The operands of an RP operator can be any legal mekso operand, including parenthesized mekso that can contain any valid syntax, whether more RP or something more conventional.
In Lojban, RP operators are always parsed with exactly two operands. What about operators which require only one operand, or more than two operands? The null operand
tu'o and the null operator
ge'a provide a simple solution. A oneoperand operator like va'a always appears in a reverse Polish context as
 tu'o va'a. The
tu'o provides the second operand, which is semantically ignored but grammatically necessary. Likewise, the threeoperand version of
gei appears in reverse Polish as
 ge'a gei, where the
ge'a effectively merges the 2nd and 3rd operands into a single operand. Here are some examples:
 Example .112:
 li fu'a ciboi muboi vu'u du li fu'a reboi tu'o va'a
Thenumber (RP!) (three, five, minus) equals thenumber (RP!) two, null, negativeof.
{{{1}}}
 Example .113:
 li cinoki'oki'o du li fu'a biboi ciboi panoboi ge'a gei
Thenumber 30commacomma equals thenumber (RP!) 8, (3, 10, nullop), exponentialnotation.
{{{1}}}
Logical and nonlogical connectives within mekso
The following cmavo are discussed in this section:
 .abu
 BY
 description
 by
 BY
 description
 cy
 BY
 description
 fe'a
 VUhU
 description
 lo'o
 LOhO
 description
As befits a logical language, Lojban has extensive provision for logical connectives within both operators and operands. Full details on logical and nonlogical connectives are provided in
Chapter ELGERROR in Template:Lch. Operands are connected in afterthought with selma'o A and in forethought with selma'o GA, just like sumti. Operators are connected in afterthought with selma'o JA and in forethought with selma'o GUhA, just like tanru components. This parallelism is no accident.
In addition, A+BO and A+KE constructs are allowed for grouping logically connected operands, and ke…ke'e is allowed for grouping logically connected operators, although there are no analogues of tanru among the operators.
Despite the large number of rules required to support this feature, it is of relatively minor importance in the mekso scheme of things. Example .114 exhibits afterthought logical connection between operands:
 Example .114:
 vei ci .a vo ve'o prenu cu klama le zarci
( Three or four ) people goto the market.
Example .115 is equivalent in meaning, but uses forethought connection:
 Example .115:
 vei ga ci gi vo ve'o prenu cu klama le zarci
( Either 3 or 4 ) people goto the market.
Note that the mekso here are being used as quantifiers. Lojban requires that any mekso other than a simple number be enclosed in parentheses when used as a quantifier. This rule prevents ambiguities that do not exist when using li.
By the way, li has an elidable terminator, lo'o, which is needed when a
li sumti is followed by a logical connective that could seem to be within the mekso. For example:
 Example .116:
 li re su'i re du li vo lo'o .onai lo nalseldjuno namcu
Thenumber two plus two equals thenumber four or else a nonknown number.
Omitting the lo'o would cause the parser to assume that another operand followed the
.onai and reject lo as an invalid operand.
Simple examples of logical connection between operators are hard to come by. A contrived example is:
 Example .117:
 li re su'i je pi'i re du li vo
Thenumber two plus and times two equals thenumber four.
{{{1}}}
The forethoughtconnection form of Example .117 is:
 Example .118:
 li re ge su'i gi pi'i re du li vo
thenumber two both plus and times two equals thenumber four.
{{{1}}}
Here is a classic example of operand logical connection:
 Example .119:
 go li .abu bi'epi'i vei xy. te'a re ve'o su'i by. bi'epi'i xy. su'i cy. du li no gi li xy. du li vei va'a by. ku'e su'i ja vu'u fe'a vei by. bi'ete'a re vu'u vo bi'epi'i .abu bi'epi'i cy. ve'o [ku'e] ve'o fe'i re bi'epi'i .abu
Ifandonlyif thenumber “a”times(“x” power two ) plus “b”times “x” plus “c” equals thenumber zero then thenumber x equals thenumber [ thenegationof( b ) plus or minus therootof (“b”power2 minus fourtimes “a”times “c” ) ] dividedby twotimes “a”.
 Iff ax ^{2} + bx + c = 0, then x = b ± √ (b ^{2} − 4ac) 2a
Note the mixture of styles in Example .119: the negation of b and the square root are represented by forethought and most of the operator precedence by prefixed
bi'e, but explicit parentheses had to be added to group the numerator properly. In addition, the square root parentheses cannot be removed here in favor of simple
fe'a and ku'e bracketing, because infix operators are present in the operand. Getting Example .119 to parse perfectly using the current parser took several tries: a more relaxed style would dispense with most of the bi'e cmavo and just let the standard precedence rules be understood.
Nonlogical connection with JOI and BIhI is also permitted between operands and between operators. One use for this construct is to connect operands with bi'o to create intervals:
 Example .120:
 li no ga'o bi'o ke'i pa
thenumber zero (inclusive) fromto (exclusive) one
[0,1)
 the numbers from zero to one, including zero but not including one
Intervals defined by a midpoint and range rather than beginning and end points can be expressed by mi'i:
 Example .121:
 li pimu ga'o mi'i ke'i pimu
thenumber 0.5 ± 0.5
which expresses the same interval as Example .120. Note that the ga'o and ke'i still refer to the endpoints, although these are now implied rather than expressed. Another way of expressing the same thing:
 Example .122:
 li pimu su'i ni'upimu bi'o ma'upimu
thenumber 0.5 plus [0.5 fromto +0.5]
Here we have the sum of a number and an interval, which produces another interval centered on the number. As Example .122 shows, nonlogical (or logical) connection of operands has higher precedence than any mekso operator.
You can also combine two operands with ce'o, the sequence connective of selma'o JOI, to make a compound subscript:
 Example .123:
 xy. xi vei by. ce'o dy. [ve'o]
“x” sub (
“b” sequence
“d”)
x<subscript>b,d</subscript>
Using Lojban resources within mekso
The following cmavo are discussed in this section:
 na'u
 NAhU
 description
 NIhE
 description
 MOhE
 description
 te'u
 TEhU
 description
One of the mekso design goals requires the ability to make use of Lojban's vocabulary resources within mekso to extend the builtin cmavo for operands and operators. There are three relevant constructs: all three share the elidable terminator te'u (which is also used to terminate vectors marked with jo'i)
The cmavo na'u makes a selbri into an operator. In general, the first place of the selbri specifies the result of the operator, and the other unfilled places specify the operands:
 Example .124:
 li na'u tanjo te'u vei pai fe'i re [ve'o] du li ci'i
Thenumber theoperator tangent (π / 2 ) = thenumber infinity.
{{{1}}}
tanjo is the gismu for “x1 is the tangent of x2”, and the na'u here makes it into an operator which is then used in forethought
The cmavo ni'e makes a selbri into an operand. The x1 place of the selbri generally represents a number, and therefore is often a
ni abstraction, since ni abstractions represent numbers. The ni'e makes that number available as a mekso operand. A common application is to make equations relating pure dimensions:
 Example .125:
 li ni'e ni clani [te'u] pi'i ni'e ni ganra [te'u] pi'i ni'e ni condi te'u du li ni'e ni canlu
Thenumber quantityof length times quantityof width times quantityof depth equals thenumber quantityof volume.
{{{1}}}
The cmavo mo'e operates similarly to
ni'e, but makes a sumti (rather than a selbri) into an operand. This construction is useful in stating equations involving dimensioned numbers:
 Example .126:
 li mo'e re ratcu su'i mo'e re ractu du li mo'e vo danlu
Thenumber two rats plus two rabbits equals thenumber four animals.
{{{1}}}
Another use is in constructing Lojbanic versions of socalled “folk quantifiers”, such as
“a pride of lions”:
 Example .127:
 mi viska vei mo'e lo'e lanzu ve'o cinfo
I see ( thetypical family )numberof lions.
 I see a pride of lions.
Other uses of mekso
The following cmavo are discussed in this section:
 me'o
 LI
 description
 nu'a
 NUhA
 description
 mai
 MAI
 description
 mo'o
 MAI
 description
 ROI
 description
So far we have seen mekso used as sumti (with li), as quantifiers (often parenthesized), and in MOI and MEMOI selbri. There are a few other minor uses of mekso within Lojban.
The cmavo me'o has the same grammatical use as li but slightly different semantics. li means “the number which is the value of the mekso ...”, whereas me'o just means “the mekso ...” So it is true that:
 Example .128:
 li re su'i re du li vo
Thenumber two plus two equals thenumber four.
{{{1}}}
but false that:
 Example .129:
 me'o re su'i re du me'o vo
Themekso two plus two equals themekso four.
 “2 + 2”=“4”
since the expressions “2 + 2” and “4” are not the same. The relationship between li and me'o is related to that between
 la djan., the person named John, and
 zo .djan., the name
“John”
The cmavo nu'a is the inverse of na'u, and allows a mekso operator to be used as a normal selbri, with the place structure:
 x1 is the result of applying (operator) to x2, x3, ...
for as many places as may be required. For example:
 Example .130:
 li ni'umu cu nu'a va'a li ma'umu
Thenumber 5 isthenegationof thenumber +5.
uses nu'a to make the operator va'a into a twoplace bridi
Used together, nu'a and na'u make it possible to ask questions about mekso operators, even though there is no specific cmavo for an operator question, nor is it grammatical to utter an operator in isolation. Consider Example .131, to which Example .132 is one correct answer:
 Example .131:
 li re na'u mo re du li vo
Thenumber two whatoperator? two equals thenumber four.
{{{1}}}
 Example .132:
 nu'a su'i
 plus
In Example .131,
 na'u mo is an operator question, because
mo is the selbri question cmavo and na'u makes the selbri into an operator. Example .132 makes the true answer su'i into a selbri (which is a legal utterance) with the inverse cmavo nu'a. Mechanically speaking, inserting Example .132 into Example .131 produces:
 Example .133:
 li re na'u nu'a su'i re du li vo
Thenumber two (theoperator theselbri plus) two equals thenumber four.
where the
 na'u nu'a cancels out, leaving a truthful bridi
Numerical free modifiers, corresponding to English
“firstly”,
“secondly”, and so on, can be created by suffixing a member of selma'o MAI to a digit string or a lerfu string. (Digit strings are compound cmavo beginning with a cmavo of selma'o PA, and containing only cmavo of PA or BY; lerfu strings begin with a cmavo of selma'o BY, and likewise contain only PA or BY cmavo.) Here are some examples:
 Example .134:
 pamai
 firstly
 Example .135:
 remai
 secondly
 Example .136:
 romai
allly
 lastly
 Example .137:
 ny.mai
 nthly
 Example .138:
 pasomo'o
 nineteenthly (higher order)
The difference between mai and mo'o is that mo'o enumerates larger subdivisions of a text. Each mo'o subdivision can then be divided into pieces and internally numbered with mai. If this chapter were translated into Lojban, each section would be numbered with mo'o. (See Section for more on these words.)
A numerical tense can be created by suffixing a digit string with
roi. This usage generates tenses corresponding to English
“once”, “twice”, and so on. This topic belongs to a detailed discussion of Lojban tenses, and is explained further in Section .
Note: the elidable terminator boi is not used between a number and a member of MAI or ROI.
Explicit operator precedence
As mentioned earlier, Lojban does provide a way for the precedences of operators to be explicitly declared, although current parsers do not understand these declarations.
The declaration is made in the form of a metalinguistic comment using
ti'o, a member of selma'o SEI.
sei, the other member of SEI, is used to insert metalinguistic comments on a bridi which give information about the discourse which the bridi comprises. The format of a ti'o declaration has not been formally established, but presumably would take the form of mentioning a mekso operator and then giving it either an absolute numerical precedence on some preestablished scale, or else specifying relative precedences between new operators and existing operators.
In future, we hope to create an improved machine parser that can understand declarations of the precedences of simple operators belonging to selma'o VUhU. Originally, all operators would have the same precedence. Declarations would have the effect of raising the specified cmavo of VUhU to higher precedence levels. Complex operators formed with na'u, ni'e, or
ma'o would remain at the standard low precedence; declarations with respect to them are for future implementation efforts. It is probable that such a parser would have a set of “commonly assumed precedences” built into it (selectable by a special ti'o declaration) that would match mathematical intuition: times higher than plus, and so on.
Miscellany
A few other points:
se can be used to convert an operator as if it were a selbri, so that its arguments are exchanged. For example:
 Example .139:
 li ci se vu'u vo du li pa
Thenumber three (inverse) minus four equals thenumber one.
 3 subtracted from 4 equals 1.
The other converters of selma'o SE can also be used on operators with more than two operands, and they can be compounded to create (probably unintelligible) operators as needed.
Members of selma'o NAhE are also legal on an operator to produce a scalar negation of it. The implication is that some other operator would apply to make the bridi true:
 Example .140:
 li ci na'e su'i vo du li pare
Thenumber 3 nonplus 4 equals thenumber 12.
 Example .141:
 li ci to'e vu'u re du li mu
Thenumber 3 oppositeofminus 2 equals thenumber 5.
The sense in which “plus” is the opposite of “minus” is not a mathematical but rather a linguistic one; negated operators are defined only loosely.
la'e and lu'e can be used on operands with the usual semantics to get the referent of or a symbol for an operand. Likewise, a member of selma'o NAhE followed by bo serves to scalarnegate an operand, implying that some other operand would make the bridi true:
 Example .142:
 li re su'i re du li na'ebo mu
Thenumber 2 plus 2 equals thenumber non5.
 2 + 2 = something other than 5.
The digits 09 have rafsi, and therefore can be used in making lujvo. Additionally, all the rafsi have CVC form and can stand alone or together as names:
 Example .143:
 la zel. poi gunta la tebes. pu nanmu
Thosenamed “Seven” who attack thatnamed “Thebes” [past] aremen.
 The Seven Against Thebes were men.
Of course, there is no guarantee that the name zel. is connected with the number rafsi: an alternative which cannot be misconstrued is:
 Example .144:
 la zemei poi gunta la tebes. pu nanmu
Thosenamedthe Sevensome who attack Thebes [past] aremen.
Certain other members of PA also have assigned rafsi: so'a,
so'e,
so'i,
so'o,
so'u,
da'a, ro, su'e,
su'o, pi, and ce'i. Furthermore, although the cmavo
fi'u does not have a rafsi as such, it is closely related to the gismu
frinu, meaning “fraction”; therefore, in a context of numeric rafsi, you can use any of the rafsi for frinu to indicate a fraction slash.
A similar convention is used for the cmavo cu'o of selma'o MOI, which is closely related to
cunso (probability); use a rafsi for cunso in order to create lujvo based on cu'o. The cmavo
mei and moi of MOI have their own rafsi, two each in fact: mem/ mei and mom/ moi respectively.
The grammar of mekso as described so far imposes a rigid distinction between operators and operands. Some flavors of mathematics (lambda calculus, algebra of functions) blur this distinction, and Lojban must have a method of doing the same. An operator can be changed into an operand with
 ni'enu'a, which transforms the operator into a matching selbri and then the selbri into an operand.
To change an operand into an operator, we use the cmavo ma'o, already introduced as a means of changing a lerfu string such as fy. into an operator. In fact, ma'o can be followed by any mekso operand, using the elidable terminator te'u if necessary.
There is a potential semantic ambiguity in
 ma'o fy. [te'u] if
fy. is already in use as a variable: it comes to mean “the function whose value is always f”. However, mathematicians do not normally use the same lerfu words or strings as both functions and variables, so this case should not arise in practice.
Four score and seven: a mekso problem
Abraham Lincoln's Gettysburg Address begins with the words
“Four score and seven years ago”. This section exhibits several different ways of saying the number
“four score and seven”. (A “score”, for those not familiar with the term, is 20; it is analogous to a “dozen” for 12.) The trivial way:
 Example .145:
 li bize
eight seven
87
Example .145 is mathematically correct, but sacrifices the spirit of the English words, which are intended to be complex and formal.
 Example .146:
 li vo pi'i reno su'i ze
four times twenty plus seven
4 × 20 + 7
Example .146 is also mathematically correct, but still misses something. “Score” is not a word for 20 in the same way that “ten” is a word for 10: it contains the implication of 20 objects. The original may be taken as short for “Four score years and seven years ago”. Thinking of a score as a twentysome rather than as 20 leads to:
 Example .147:
 li mo'e voboi renomei te'u su'i ze
thenumberof four twentysomes plus seven
In Example .147,
 voboi renomei is a sumti signifying four things each of which are groups of twenty; the
mo'e and
te'u then make this sumti into a number in order to allow it to be the operand of su'i.
Another approach is to think of “score” as setting a representation base. There are remnants of base20 arithmetic in some languages, notably French, in which 87 is
“quatrevingtsept”, literally “fourtwentiesseven”. (This fact makes the Gettysburg Address hard to translate into French!) If
“score” is the representation base, then we have:
 Example .148:
 li vo pi'e ze ju'u reno
four ; seven base 20
47
Overall, Example .147 probably captures the flavor of the English best. Example .145 and Example .146 are too simple, and Example .148 is too tricky. Nevertheless, all four examples are good Lojban. Pedagogically, these examples illustrate the richness of lojbau mekso: anything that can be said at all, can probably be said in more than one way.
mekso selma'o summary
Except as noted, each selma'o has only one cmavo.
 BOI
 description
 BY
 description
 FUhA
 description
 GOhA
 description
 JOhI
 description
 KUhE
 description
 LI
 description
 MAhO
 description
 MOI
 description
 MOhE
 description
 NAhU
 description
 NIhE
 description
 NUhA
 description
 PA
 description
 PEhO
 description
 TEhU
 description
 VEI
 description
 VEhO
 description
 VUhU
 description
 XI
 description
Complete table of VUhU cmavo, with operand structures
The operand structures specify what various operands (labeled a, b, c, ...) mean. The implied context is forethought, since only forethought operators can have a variable number of operands; however, the same rules apply to infix and RP uses of VUhU.
 su'i
 description
<description role="math">(((a + b) + c) + ...)</description>
 pi'i
 description
<description role="math">(((a × b) × c) × ...)</description>
 vu'u
 description
<description role="math">(((a − b) − c) − ...)</description>
 fe'i
 description
<description role="math">(((a / b) / c) / ...)</description>
 ju'u
 description
<description role="math">numeral string a interpreted in the base b</description>
 pa'i
 description
<description role="math">the ratio of a to b a:b</description>
 fa'i
 description
<description role="math">1 / a</description>
 gei
 description
<description role="math">b × (c [default 10] to the a power)</description>
 ge'a
 description
<description role="math">(no operands)</description>
 de'o
 description
<description role="math">log a to base b (default 10 or e as appropriate)</description>
 te'a
 description
<description role="math">a to the b power</description>
 fe'a
 description
<description role="math">b^{th} root of a (default square root: b = 2)</description>
 cu'a
 description
<description role="math"></description>
 ne'o
 description
<description role="math">a!</description>
 pi'a
 description
<description role="math">(all operands are row vectors)</description>
 sa'i
 description
<description role="math">(all operands are column vectors)</description>
 ri'o
 description
<description role="math">integral of a with respect to b over range c</description>
 sa'o
 description
<description role="math">derivative of a with respect to b of degree c (default 1)</description>
 fu'u
 description
<description role="math">(variable)</description>
 si'i
 description
<description role="math">summation of a using variable b over range c</description>
 va'a
 description
<description role="math">a</description>
 re'a
 description
<description role="math">a^{*}</description>
Complete table of PA cmavo: digits, punctuation, and other numbers.
<title> Decimal digits:==
non <description role="math">0</description>
pav <description role="math">1</description>
rel <description role="math">2</description>
cib <description role="math">3</description>
von <description role="math">4</description>
mum <description role="math">5</description>
xav <description role="math">6</description>
zel <description role="math">7</description>
biv <description role="math">8</description>
soz <description role="math">9</description>
<title> Hexadecimal digits:==
<description role="math">A/10</description>
<description role="math">B/11</description>
<description role="math">C/12</description>
<description role="math">D/13</description>
<description role="math">E/14</description>
<description role="math">F/15</description>
<title> Special numbers:==
<description role="math">π</description>
 ka'o
 description
 te'o
 description
 ci'i
 description
<title> Number punctuation:==
piz
 description
cez
 description
fi'u (from frinu; see Section .19)
 description
 pi'e
 description
 ma'u
 description
 ni'u
 description
 ki'o
 description
 ra'e
 description
 ji'i
 description
 ka'o
 description
<title> Indefinite numbers:==
 ro
 description
rol
soj
 description
sop
 description
 so'i
 description
sor so'i
sos
 description
sot
 description
daz
 description
<title>Subjective numbers:==
 rau
 description
 du'e
 description
 mo'a
 description
<title>Miscellaneous:==
 xo
 description
 tu'o
 description
Table of MOI cmavo, with associated rafsi and place structures
mem mei <description role="placestructure">x1 is a mass formed from a set x2 of n members, one or more of which is/are x3, [measured relative to the set x4/by standard x4]</description>
mom moi <description role="placestructure">x1 is the (n)th member of set x2 when ordered by rule x3 [by standard x4]</description>
<rafsi/> <description role="placestructure">x1 is an (n)th portion of mass x2 [by standard x3]</description>
cu'o (borrowed from cunso; see Section .19) <description role="placestructure">event x1 has probability (n) of occurring under conditions x2 [by standard x3]</description>
<rafsi/> <description role="placestructure">x1 is at scale position (n) on the scale x2 [by standard x3]</description>